It is also possible to see the contents of each and every character set through an admin menu page, which displays each character set just as it would appear in the edit widget. (And I customized the look of the character sets so they look like keys on a keyboard.) The following is how the Common character set appears.
As you can see, the character sets are broken down into contiguous subsets according to their contents, in this case Text, Punctuation, Quote Marks, Signs and Symbols and Mathematics, to make locating the desired character easier. Furthermore, when the cursor hovers over a character, a title box will be displayed showing the Unicode designation and a description of the character.